Few-view CT image reconstruction system

ABSTRACT

A system for few-view computed tomography (CT) image reconstruction is described. The system includes a preprocessing module, a first generator network, and a discriminator network. The preprocessing module is configured to apply a ramp filter to an input sinogram to yield a filtered sinogram. The first generator network is configured to receive the filtered sinogram, to learn a filtered back-projection operation and to provide a first reconstructed image as output. The first reconstructed image corresponds to the input sinogram. The discriminator network is configured to determine whether a received image corresponds to the first reconstructed image or a corresponding ground truth image. The generator network and the discriminator network correspond to a Wasserstein generative adversarial network (WGAN). The WGAN is optimized using an objective function based, at least in part, on a Wasserstein distance and based, at least in part, on a gradient penalty.

CROSS REFERENCE TO RELATED APPLICATION(S)

This application claims the benefit of U.S. Provisional Application No.62/899,517, filed Sep. 12, 2019, and U.S. Provisional Application No.63/077,745, filed Sep. 14, 2020, which are incorporated by reference asif disclosed herein in their entirety.

GOVERNMENT LICENSE RIGHTS

This invention was made with government support under award numbersCA233888 and CA237267, awarded by the National Institutes of Health(NIH), and under award number EB026646, awarded by the NationalInstitutes of Health (NIH). The government has certain rights in theinvention.

FIELD

The present disclosure relates to few-view CT (computed tomography)image reconstruction.

BACKGROUND

X-ray computed tomography (CT) is a popular medical imaging method forscreening, diagnosis, and image guided intervention. Although CT bringsoverwhelming healthcare benefits to patients, it may potentiallyincrease cancer risk due to the involved ionizing radiation. Low-dose CTand few-view CT result in a reduced exposure to the ionizing radiationbut typically at a cost of reduced image quality.

SUMMARY

In an embodiment, there is provided a system for few-view computedtomography (CT) image reconstruction. The system includes apreprocessing module, a first generator network, and a discriminatornetwork. The preprocessing module is configured to apply a ramp filterto an input sinogram to yield a filtered sinogram. The first generatornetwork is configured to receive the filtered sinogram, to learn afiltered back-projection operation and to provide a first reconstructedimage as output. The first reconstructed image corresponds to the inputsinogram. The discriminator network is configured to determine whether areceived image corresponds to the first reconstructed image or acorresponding ground truth image. The first generator network and thediscriminator network correspond to a Wasserstein generative adversarialnetwork (WGAN). The WGAN is optimized using an objective function based,at least in part, on a Wasserstein distance and based, at least in part,on a gradient penalty.

In some embodiments, the system further includes a second generatornetwork. The second generator network is configured to receive aconcatenation of the first reconstructed image and a filteredback-projection of the input sinogram. The second generator network isfurther configured to provide a second reconstructed image. Thediscriminator network is further configured to determine whether thereceived image corresponds to the second reconstructed image.

In some embodiments of the system, the first generator network isconfigured to learn the filtered back-projection operation in apoint-wise manner.

In some embodiments of the system, the first generator network includesa filtration portion, a back-projection portion, and a refinementportion.

In some embodiments of the system, the WGAN is trained, initially, usingimage data from an image database including a plurality of images.

In some embodiments of the system, the first generator network isconfigured to reconstruct the first reconstructed image usingO(C×N×N_(v)) parameters, where N is a dimension of the firstreconstructed image, N_(v) is a number of projections and C is anadjustable hyper-parameter in the range of 1 to N.

In some embodiments of the system, the second generator networkcorresponds to a refinement portion.

In an embodiment, there is provided a method for few-view computedtomography (CT) image reconstruction. The method includes applying, by apreprocessing module, a ramp filter to an input sinogram to yield afiltered sinogram; receiving, by a first generator network, the filteredsinogram; learning, by the first generator network, a filteredback-projection operation; and providing, by the first generatornetwork, a first reconstructed image as output. The first reconstructedimage corresponds to the input sinogram. The method further includesdetermining, by a discriminator network, whether a received imagecorresponds to the first reconstructed image or a corresponding groundtruth image. The first generator network and the discriminator networkcorrespond to a Wasserstein generative adversarial network (WGAN). TheWGAN is optimized using an objective function based, at least in part,on a Wasserstein distance and based, at least in part, on a gradientpenalty.

In some embodiments, the method further includes receiving, by a secondgenerator network, a concatenation of the first reconstructed image anda filtered back-projection of the input sinogram; providing, by thesecond generator network, a second reconstructed image; and determining,by the discriminator network, whether the received image corresponds tothe second reconstructed image.

In some embodiments of the method, the first generator network isconfigured to learn the filtered back-projection operation in apoint-wise manner.

In some embodiments of the method, the first generator network includesa filtration portion, a back-projection portion, and a refinementportion.

In some embodiments, the method further includes learning, by the firstgenerator network, an initial filtered back-projection operation usingimage data from an image database including a plurality of images.

In some embodiments of the method, the first generator network isconfigured to reconstruct the first reconstructed image usingO(C×N×N_(v)) parameters, where N is a dimension of the firstreconstructed image, N_(v) is a number of projections and C is anadjustable hyper-parameter in the range of 1 to N.

In some embodiments, the method further includes receiving, by afiltered back projection module, the input sinogram and providing, bythe filtered back projection module, the filtered back-projection of theinput sinogram.

In an embodiment, there is provided a computer readable storage device.The device has stored thereon instructions configured for few-viewcomputed tomography (CT) image reconstruction. The instructions thatwhen executed by one or more processors result in the followingoperations including: applying a ramp filter to an input sinogram toyield a filtered sinogram; receiving the filtered sinogram; learning afiltered back-projection operation; providing a first reconstructedimage as output, the first reconstructed image corresponding to theinput sinogram; and determining whether a received image corresponds tothe first reconstructed image or a corresponding ground truth image, theoperations corresponding to a Wasserstein generative adversarial network(WGAN). The WGAN is optimized using an objective function based, atleast in part, on a Wasserstein distance and based, at least in part, ona gradient penalty.

In some embodiments of the device, the operations further includereceiving a concatenation of the first reconstructed image and afiltered back-projection of the input sinogram; providing a secondreconstructed image; and determining whether the received imagecorresponds to the second reconstructed image.

In some embodiments of the device, the filtered back-projectionoperation is learned in a point-wise manner.

In some embodiments of the device, the operations further includelearning an initial filtered back-projection operation using image datafrom an image database including a plurality of images.

In some embodiments of the device, the first reconstructed image isreconstructed using O(C×N×N_(v)) parameters, where N is a dimension ofthe first reconstructed image, N_(v) is a number of projections and C isan adjustable hyper-parameter in the range of 1 to N.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings show embodiments of the disclosed subject matter for thepurpose of illustrating features and advantages of the disclosed subjectmatter. However, it should be understood that the present application isnot limited to the precise arrangements and instrumentalities shown inthe drawings, wherein:

FIG. 1 illustrates a functional block diagram of a system that includesa deep learning CT image reconstruction system consistent with severalembodiments of the present disclosure;

FIG. 2 illustrates a functional block diagram of a system that includesa dual network architecture (DNA) CT image reconstruction systemconsistent with several embodiments of the present disclosure;

FIG. 3 is a flow chart of deep learning CT image reconstruction systemtraining operations according to various embodiments of the presentdisclosure; and

FIG. 4 is a flow chart of dual network architecture (DNA) CT imagereconstruction system training operations according to variousembodiments of the present disclosure.

DETAILED DESCRIPTION

Current commercial CT scanners typically use one or two x-ray sourcesthat are mounted on a rotating gantry to take hundreds of projections atdifferent angles around a patient's body. The rotating mechanism ismassive and consumes substantial energy related to a net angularmomentum generated during the rotation. Thus, outside major hospitals,current commercial CT scanners are largely inaccessible due to theirsize, weight and expense. Few-view CT may be implemented in amechanically stationary scanner thus avoiding the rotating mechanism andassociated power consumption.

The Nyquist sampling theorem provides a lower bound on an amount of dataused for image reconstruction. For example, when sufficient (i.e., abovethe Nyquist limit) projection data are acquired, analytic methods suchas filtered back-projection (FBP) may provide relatively high quality CTimage reconstruction. In few-view CT, due in part to under-sampled data,streak artifacts may be introduced in analytically reconstructed imagesbecause of incomplete projection data. Iterative techniques mayincorporate prior knowledge in the image reconstruction but can berelatively time-consuming and may not produce satisfying results in somecases.

Generally, the present disclosure relates to a few-view CT imagereconstruction system. In an embodiment, the few-view CT imagereconstruction system corresponds to a deep efficient end-to-endreconstruction (DEER) network for few-view CT image reconstruction. Inanother embodiment, the few-view CT image reconstruction systemcorresponds to a dual network architecture (DNA) CT image reconstructionsystem. A method and/or system consistent with the present disclosuremay be configured to receive CT scanner projection data (i.e.,sinograms) and to generate a corresponding image. A system may includeat least one generator network and a discriminator network configured asa generative adversarial neural network (GAN). The generator network(s)and the discriminator network correspond to artificial neural networks.In an embodiment, the generator network(s) and the discriminator networkmay correspond to convolutional neural networks. The generatornetwork(s) and discriminator network may be trained, adversarially, aswill be described in more detail below. The trained generator network(s)may then be configured to receive filtered few view projection data andto provide a reconstructed image as output.

In an embodiment, at least one generator network may correspond to aback projection network (BPN). The BPN may be configured to reconstructa CT image directly from raw (i.e., sinogram) data using, for example,O(C×N×N_(v)) parameters. N corresponds to dimension of reconstructedimage and N_(v) corresponds to number of projections. C is an adjustablehyper-parameter and is in the range of 1 to N. A BPN consistent with thepresent disclosure may thus be trainable on one consumer-level GPU(graphics processing unit). However, this disclosure is not limited inthis regard. The BPN, similar to filtered back projection (FBP), isconfigured to learn a refined filtration back-projection process forreconstructing images directly from sinograms. For X-ray CT, each pointin the sinogram domain relates to pixels/voxels on an X-ray path througha field of view. Thus, a plurality of line integrals acquired by aplurality of different detectors at particular angle are not related toeach other. With this intuition, the reconstruction process of BPN islearned in a point-wise manner that facilitates constraining a memoryburden.

In some embodiments, the generator network may be pre-trained usingnatural images from a publicly available image database, e.g., ImageNet.The generator network may then be refined using actual patient data.Advantageously, the complexity of natural images may facilitate learningthe back-projection process.

FIG. 1 illustrates a functional block diagram of a system 100 thatincludes a deep learning CT image reconstruction system 102, consistentwith several embodiments of the present disclosure. CT imagereconstruction system 102 includes elements configured to implementtraining of a back projection network (BPN), as will be described inmore detail below. System 100 further includes a computing device 104.Computing device 104 is configured to perform the operations of deeplearning CT image reconstruction system 102.

The computing device 104 may include, but is not limited to, a server, aworkstation computer, a desktop computer, a laptop computer, a tabletcomputer, an ultraportable computer, an ultramobile computer, a netbookcomputer and/or a subnotebook computer, etc. Computing device 104includes a processor 110, a memory 112, input/output (I/O) circuitry114, a user interface (UI) 116, and storage 118.

CT image reconstruction system 102 includes a training module 120, atraining data store 122, a preprocessing module 124, a generator network126, and a discriminator network 128. Generator network 126 includes afiltration portion 126-1, a back-projection portion 126-2 and arefinement portion 126-3. Generator network 126, after training,corresponds to a BPN. As used herein, the terms “generator network” and“generative network” are used interchangeably.

Processor 110 may include one or more processing units and is configuredto perform operations of system 100, e.g., operations of training module120, preprocessing module 124, generator network 126, and discriminatornetwork 128. Memory 112 may be configured to store data associated withtraining module 120, preprocessing module 124, generator network 126,and discriminator network 128, and/or training data store 122. I/Ocircuitry 114 may be configured to communicate wired and/or wirelesslywith a source of projection data and/or a recipient of a correspondinggenerated image. UI 116 may include a user input device (e.g., keyboard,mouse, microphone, touch sensitive display, etc.) and/or a user outputdevice, e.g., a display. Storage 118 is configured to store at least aportion of training data store 122. Training data store 122 isconfigured to store training data including, but not limited to, one ormore objective functions 140, one or more training data sets 142,generator parameters 146 and discriminator parameters 148.

Training module 120 is configured to manage training operations ofgenerator network 126 (and discriminator network 128). Training module120 may thus be configured to provide training projection data topreprocessing module 124 and ground truth image data to discriminatornetwork 128. The training projection data and ground truth image datamay be stored, for example, in training data store 122 as training datasets 142. Training module 120 may be further configured to provide anobjective function, e.g., objective function 140, to discriminatornetwork 128 and to receive a decision from discriminator network.Training module 120 may be further configured to provide, adjust and/orreceive generator parameters 127 and/or discriminator parameters 129during training operations. Such parameters may include, for example,neural network weights. Generator parameters may be stored in trainingdata store as generator parameters 146 and discriminator parameters maybe stored in training data store as discriminator parameters 148. Aftertraining, i.e., during normal operations, the generator parameters maybe set, CT image reconstruction system 102 may be configured to receiveprojection data in (corresponding to an actual CT sinogram) and may beconfigured to provide a corresponding generated image 121 as generatedimage output.

CT image reconstruction may be expressed as:I _(FV) =R ⁻¹(S _(SV))  (1)where I_(FV)∈

^(w×w) is an object image with dimension w×w, S_(SV)∈

^(v×w) is the sinogram with dimension v×w and R⁻¹ corresponds to aninverse radon transform (e.g., filtered back projection (FBP)) in aninstance where sufficient two dimensional (2D) projection data isavailable. When sufficient 2D projection data is available, CT imagereconstruction can be reduced to solving a system of linear equations.If the number of linear equations is less than the number of unknownpixels as in the few-view CT setting, the image reconstruction is anunderdetermined problem. Deep learning (DL) may be utilized to extractfeatures of raw data for image reconstruction. With a deep neuralnetwork, as described herein, training data corresponds to priorknowledge configured to establish a relationship between a sinogram andthe corresponding CT image. Thus, a trained deep neural network may beconfigured to efficiently solving this undetermined problem.

In operation, CT image reconstruction system 102 is configured as aWasserstein Generative Adversarial Network (WGAN) to optimize (i.e.,train) generator network 126. After optimization, generator network 126may correspond to a back projection network (BPN). The BPN 126 isconfigured to receive preprocessed, as described herein, few view CTprojection data, and to reconstruct a corresponding CT image.

A WGAN generally includes a generator network, e.g., generator network126, and a discriminator network, e.g., discriminator network 128. Thegenerator network 126 aims at reconstructing images directly from abatch of few-view sinograms. The discriminator network 128 is configuredto receive generated image data 121 from generator network 126 or groundtruth image data from, e.g., training module 120, and intends todistinguish whether an image is real (i.e., ground truth) or fake (fromgenerator network 126). Both networks 126, 128 are configured to beoptimized during the training process. If an optimized discriminatornetwork 128 can hardly distinguish fake images from real images, thengenerator network 126 can fool discriminator network 128 which is thegoal of WGAN. In other words, if discriminator network 128 is unable todistinguish between a generated image from the generator network 126 anda ground truth image, the generator network 126 has been optimized,i.e., is trained. The discriminator network 128 may facilitate improvinga texture of the final image and reduce occurrence of over-smoothing.

WGAN is configured to replace a cross-entropy loss function of anon-Wasserstein generative adversarial network (GAN) with theWasserstein distance. The Wasserstein distance is configured to improvethe training stability during the training process compared to the GAN.In an embodiment, an objective function used during training includesthe Wasserstein distance as well as a gradient penalty term. Theobjective function of the discriminator network 128 may be written as:

$\begin{matrix}{\min\limits_{\theta_{G}}\max\limits_{\theta_{D}}\left\{ {{{\mathbb{E}}_{S_{SV}}\left\lbrack {D\left( {G\left( S_{SV} \right)} \right)} \right\rbrack} - {{\mathbb{E}}_{I_{FV}}\left\lbrack {D\left( I_{FV} \right)} \right\rbrack} + {{\lambda\mathbb{E}}_{\overset{\_}{I}}\left\lbrack \left( {{{\nabla\left( \overset{\_}{I} \right)}}_{2} - 1} \right)^{2} \right\rbrack}} \right\}} & \left( {2A} \right)\end{matrix}$where D corresponds to operation of the discriminator network 128, Gcorresponds to operation of the generator network 126, S_(SV) and I_(FV)represent sparse-view sinograms and ground-truth images, respectively.Terms of the form

_(a)[b] in Eq. 2A denote an expectation of b as a function of a. θ_(G)and θ_(D) represent the trainable parameters of the generator network126 and the discriminator network 128, respectively.Ī=α·I_(FV)+(1−α)·G(S_(SV)). α is uniformly sampled from the interval[0,1]. In other words, Ī represents images between fake and real images.∇(Ī) denotes the gradient of D with respect to Ī. λ is a parameter usedto balance the Wasserstein distance term and gradient penalty term. Thegenerator network 126 and the discriminator network 128 (e.g., thegenerator parameters and the discriminator parameters) may be updatediteratively.

The input to the BPN 126 is a batch of few-view sinograms. According toFourier slice theorem, low-frequency information is sampled denser thanhigh-frequency information. It may be appreciated that performingback-projection directly on the batch of few-view sinograms may resultin blurry reconstructed images. Preprocessing module 124 is configuredto implement a ramp filter to filter received projection data (i.e.,sinogram) to avoid this blurry issue. The ramp filter operation may beperformed on sinograms in the Fourier domain as multiplication. Thefilter length may be set as twice the length of the sinogram.Theoretically, the length of the filter is infinitely long for abandlimited signal, but it is not practical in reality. Since values ofthe filter outside twice the length of sinograms are generally at ornear zero, filter length is set as twice the length of sinograms. Thefiltered sinograms, i.e., output of the preprocessing module 124, maythen be provided to the generator network 126.

Generator network 126 is configured to learn a revised filtrationback-projection operation and output reconstructed images, i.e.,generated image 121. Generator network 126 includes three components: afiltration portion 126-1, a back-projection portion 126-2 and arefinement portion 126-3.

In filtration portion 126-1, a plurality of one dimensional (1-D)convolutional layers are used to learn small variance to filteredsinograms. Because the filtration portion 126-1 is a multi-layer CNN,different layers can learn different parts of the filter. In onenonlimiting example, the 1-D convolutional window may be set as onequarter the length of the sinograms. The length of the 1-D convolutionalwindow is configured to reduce computational burden. The idea ofresidual connection may be used to preserve high-resolution informationand prevent gradient from vanishing. Inspired by the ResNeXt structure,in one nonlimiting example, a cardinality of the convolutional layersmay be 3. It may be appreciated that increasing the cardinality of thenetwork may be more effective than increasing the depth or width of thenetwork when the network capacity is increased. In BPN 126, the value ofthe cardinality may correspond to a number of branches.

The learned sinograms from the filtration portion 126-1 may then beprovided to the back-projection portion 126-2. It may be appreciatedthat each point in the sinogram relates to pixel values on the x-raypath through corresponding object image and any other pixels do notcontribute to the point. Thus, the reconstruction process may be learnedin a point-wise manner using a point-wise fully-connected layer. Thus,the generator network 126, by learning in a point-wise manner, may learnthe back-projection process with relatively fewer parameters compared toother methods. Learning with relatively fewer parameters may utilizerelatively fewer memory resources. In other words, for a sinogram withdimension N_(v)×N, there is a total of N_(v)×N relatively smallfully-connected layers in the method, as described herein. Therespective input to each of these relatively small fully-connectedlayers is a single point in the sinogram and the output is a line withdimension N×1. After this point-wise fully-connected layer, rotation andsummation may be applied to simulate FBP, and to put all the learnedlines to their appropriate positions. Bilinear interpolation may be usedfor rotating images and maintaining the rotated image on a Cartesiangrid.

This network design is configured to allow the corresponding neuralnetwork to learn the reconstruction process using N parameters. In somesituations, due to the relative complexity of medical images andincomplete projection data (due to the few-view input data), Nparameters may not be sufficient for learning relatively high-qualityimages. Thus, in an embodiment, the number of parameters may beincreased to O(C×N×N_(v)) in this point-wise fully-connected layer to byincreasing the number of branches to C (an adjustable hyper-parameter).The increase in number of parameters is further supported by using adifferent set of parameters for different angles in order to compensatethe negative effect introduced by bilinear interpolation. An amount ofbias terms in this point-wise fully-connected layer is the same as theamount of weights in order to learn fine details in medical images. Biasterms are added along the detector direction. Then, there is one 2-Dconvolutional layer with 3×3 kernel and stride 1 configured to combineall the learned mappings from sinogram domain to image domain. It shouldbe noted that by learning in this point-wise manner, each point in thesinogram becomes a training sample instead of a whole sinogram and inorder to reduce training time, a plurality of fully-connected layers maybe implemented together by one piece-wise multiplication.

Images reconstructed in the back-projection portion 126-2 may then beprovided to the last portion of generator network 126, i.e., therefinement portion 126-3. Refinement portion 126-3 may be configured toremove remaining artifacts. For example, the refinement portion 126-3may correspond to a U-net, including conveying paths, and may be builtwith ResNeXt structure. The conveying paths are configured to copy earlyfeature maps and reuse them as part of the input to later layers.Concatenation is used to combine early and later feature maps along thechannel dimension. The generator network 126 may then be configured topreserve high-resolution features. Each layer in the U-net may befollowed by a rectified linear unit (ReLU). 3×3 kernels may be used inboth convolutional and transpose-convolutional layers. A stride of 2 maybe used for down-sampling and up-sampling layers and stride of 1 may beused for all other layers. In order to maintain the tensor's size,zero-padding is used.

The discriminator network 128 is configured to receive input from eithergenerator network 126 (i.e., generated image 121) or the ground-truthdataset (e.g., ground truth image data from training module 120). Asdescribed herein, the discriminator network 128 may be configured todistinguish whether the input is real or fake. In one nonlimitingexample, the discriminator network 128 may contain 6 convolutionallayers with 64, 64, 128, 128, 256, 256 filters, respectively, andfollowed by 2 fully-connected layers with number of neurons 1024 and 1,respectively. A leaky ReLU activation function may be used after eachlayer with a slope of 0.2, for example, in the negative part. Aconvolutional window of 3×3 and zero-padding may be used for allconvolutional layers. Stride may be equal to 1 for odd layers and 2 foreven layers.

Generally, the objective function used for optimizing a generatornetwork may include one or more of mean square error (MSE), adversarialloss and structural similarity index (SSIM). MSE may effectivelysuppress back-ground noise, but may result in over-smoothed images.Generally, MSE may not be sensitive to image texture. MSE generallyassumes background noise is white Gaussian noise that is independent oflocal image features. The formula of MSE loss may be expressed as:

$\begin{matrix}{L_{2} = {\frac{1}{N_{b} \cdot W \cdot H}{\sum\limits_{i = 1}^{N_{b}}{{Y_{i} - X_{i}}}_{2}^{2}}}} & (3)\end{matrix}$where N_(b), W and H correspond to the number of batches, image widthand image height respectively. Y_(i) and X_(i) represent ground-truthimage and image reconstructed by generator network 126, respectively. Inorder to compensate the disadvantages of MSE and acquire visually betterimages, SSIM is introduced in the objective function. SSIM aims tomeasure structural similarity between two images. In one nonlimitingexample, the convolution window used to measure SSIM is set as 11×11.The SSIM formula is expressed as:

$\begin{matrix}{{{SSIM}\left( {Y,X} \right)} = \frac{\left( {{2\mu_{Y}\mu_{X}} + C_{1}} \right)\left( {{2\sigma_{YX}} + C_{2}} \right)}{\left( {{\mu_{Y}}^{2} + {\mu_{X}}^{2} + C_{1}} \right)\left( {{\sigma_{Y}}^{2} + {\sigma_{X}}^{2} + C_{2}} \right)}} & (4)\end{matrix}$where C₁=(K₁·R)² and C₂=(K₂·R)² are constants used to stabilize theformula if the denominator is small. R stands for the dynamic range ofpixel values and, in one nonlimiting example, K₁=0.01 and K₂=0.03.μ_(Y), μ_(X), σ_(Y) ², σ_(X) ², and σ_(YX) are the means of Y and X,variances of Y and X and the covariance between Y and X, respectively.The structural loss may then be expressed as:L _(sl)=1−SSIM(Y,X)  (5)The adversarial learning technique used in BPN aims to help generatornetwork 126 produce sharp images that are indistinguishable by thediscriminator network 128. Referring to Eq. 1, adversarial loss may bewritten as:L _(al)=−

_(S) _(SV) [D(G(S _(SV)))]  (6)The overall objective function of the generator network 126 may then bewritten as:L _(G)=λ_(Q) ·L _(al)+λ_(P) ·L _(sl) +L ₂  (7A)where λ_(Q) and λ_(P) are hyper-parameters used to balance differentloss functions.

Thus, a deep efficient end-to-end reconstruction (DEER) network forfew-view CT image reconstruction system, consistent with the presentdisclosure, may include a generator network and a discriminator network.The generator network and discriminator network may be trained,adversarially, using a WGAN framework, as described herein. The DEERnetwork for few-view CT image reconstruction system may then beconfigured to receive CT scanner projection data (i.e., sinograms), tofilter the received projection data and to generate a correspondingimage. In some embodiments, the generator network and discriminatornetwork may be pre-trained using, for example, ImageNet data. The CTimage reconstruction process of the BPN network is learned in apoint-wise manner that facilitates constraining a memory burden.

FIG. 2 illustrates a functional block diagram of system 200 thatincludes a dual network architecture (DNA) CT image reconstructionsystem 202 consistent with several embodiments of the presentdisclosure. DNA system 202 includes elements configured to implementtraining generator network(s) and a discriminator network, as will bedescribed in more detail below. System 200 further includes computingdevice 104, as described herein. Computing device 104 is configured toperform the operations of dual network CT image reconstruction system202. Storage 118 may be configured to store at least a portion oftraining data store 222, as described herein.

It may be appreciated that DNA CT image reconstruction system 202 has atleast some elements and features in common with CT image reconstructionsystem 102 of FIG. 1 . In the interest of descriptive efficiency, thecommon elements and features will be only briefly described, withreference provided to the description herein related to the CT imagereconstruction system 102 of FIG. 1 .

CT image reconstruction system 202 includes a training module 220, atraining data store 222, a preprocessing module 224, a filtered backprojection (FBP) module 226, a first generator network (Gen 1) 228, anintermediate processing module 230, a second generator network (Gen 2)232, and a discriminator network 234. Training data store 222 isconfigured to store training data including, but not limited to, one ormore objective function(s) 240, one or more training data sets 242,first generator (Gen 1) parameters 244, second generator (Gen 2)parameters 246, and discriminator parameters 248.

The preprocessing module 224 corresponds to preprocessing module 124 ofFIG. 1 . The first generator (Gen 1) network 228 corresponds to thegenerator network 126 of FIG. 1 . Similar to the deep learning CT imagereconstruction system 102 of FIG. 1 , DNA CT image reconstruction system202 is configured to receive CT scanner projection data (i.e.,sinograms) and to generate (i.e., reconstruct) a corresponding image(Final output image 233). The DNA CT image reconstruction system 202 maybe trained, adversarially, as described herein. A subsystem thatincludes preprocessing module 224, trained generator networks 228, 232and intermediate processing module 230, may then be configured toreceive filtered projection data and to provide a reconstructed image asoutput.

DNA CT image reconstruction system 202 includes two generator networks:Gen 1 network 228 and Gen 2 network 232. As used herein, the terms “G1”,“Gen 1” and “Gen 1 network” are used interchangeably and all refer toGen 1 network 228 of FIG. 2 . As used herein, the terms “G2”, “Gen 2”and “Gen 2 network” are used interchangeably and all refer to Gen 2network 232 of FIG. 2 . Training module 220 is configured to managetraining operations of generator networks 228 and 232 and discriminatornetwork 234, similar to training module 120.

Training module 220 may thus be configured to provide trainingprojection data (i.e., input sinogram) to preprocessing module 224 andFBP module 226. Training module 220 may be further configured to provideground truth image data to discriminator network 234. The trainingprojection data and ground truth image data may be stored, for example,in training data store 222 as training data sets 242. Training module220 may be further configured to provide an objective function, e.g.,objective function 240, to discriminator network 234 and to receive adecision from discriminator network. Training module 220 may be furtherconfigured to provide, adjust and/or receive Gen 1 parameters 243, Gen 2parameters 245, and/or discriminator parameters 247 during trainingoperations. Such parameters may include, for example, neural networkweights. Gen 1 parameters may be stored in training data store as Gen 1parameters 244, Gen 2 parameters may be stored in training data store asGen 2 parameters 246, and discriminator parameters may be stored intraining data store as discriminator parameters 248. After training,i.e., during normal operations, the Gen 1 and Gen 2 parameters may beset, CT image reconstruction system 202 may be configured to receiveprojection data in (corresponding to an actual CT sinogram) and may beconfigured to provide a corresponding generated image as final outputimage 233.

In operation, preprocessing module 224 and FBP module 226 are configuredto receive training projection data (e.g., a batch of few-viewsinograms) from, e.g., training module 220. Preprocessing module 224 isconfigured to filter the few-view sinograms to yield filtered few viewsinograms 225. In one nonlimiting example, the filter length may betwice the length of the sinogram. The filtering corresponds to a rampfilter applied to the sinograms in the Fourier domain, as describedherein.

The filtered few-view sinograms 225 may then be provided to Gen 1network 228. Gen 1 network 228 corresponds to generator network 126 ofFIG. 1 . Gen 1 network 228 is configured to operate on the filteredfew-view sinograms (i.e., to learn a filtered back projection technique)to produce an intermediate output 229. The intermediate output 229 maycorrespond to reconstructed image that may then be provided to theintermediate processing module 230. The FBP module 226 is configured toperform filtered back-projection on the received training projectiondata (e.g., a batch of few-view sinograms) and to provide an FBP result227 to the intermediate processing module 230. The intermediateprocessing module 230 is configured to concatenate the intermediateoutput 229 with the FBP result 227 and to provide the concatenatedresult 231 to the Gen 2 network 232. The Gen 2 network 232 is configuredto operate on the concatenated result 231 (e.g., to optimize theconcatenated result) to produce a final output image 233. Theintermediate output 229 and the final output image 233 may be furtherprovided to the discriminator network 234. The discriminator network 234is further configured to receive ground truth image data and to provideat least one decision indicator to, for example, training module 220.

Similar to generator network 126 of FIG. 1 , the Gen 1 network 228 mayinclude three portions: filtration 228-1, back-projection 228-2, andrefinement 228-3. The filtration portion 228-1 may correspond to amulti-layer CNN. In the filtration part 228-1, 1-D convolutional layersare used to produce filtered data. In one nonlimiting example, filterlength of filtration portion 228-1 may be set to twice a length of aprojection vector. It may be appreciated that the length of theprojection vector may be shortened. Since the filtration is done througha multi-layer CNN, different layers can learn different parts of thefilter. In one nonlimiting example, the 1-D convolutional window may beempirically set as one quarter the length of the projection vector toreduce the computational burden. Residual connections may be used topreserve high-resolution information and to prevent gradient fromvanishing.

The learned sinogram from the filtration portion 228-1 may then beprovided to the back-projection portion 228-2. The back-projectionportion 228-2 is configured to perform back-projection operations on thereceived learned sinogram. Operation of the back-projection portion228-2 is inspired by the following intuition: every point in thefiltered projection vector only relates to pixel values on the x-raypath through the corresponding object image and any other data points inthis vector contribute nothing to the pixels on this x-ray path. As isknown a single fully-connected layer can be implemented to learn themapping from the sinogram domain to the image domain, but relies onrelatively large matrix multiplications in this layer that may taxmemory. To reduce the memory burden, DNA CT image reconstruction system202 (e.g., Gen 1 network 228) is configured to learn the reconstructionprocess in a point-wise manner using a point-wise fully-connected layer.Back-projection portion 228-2 may then learn the back-projectionprocess. The input to the point-wise fully-connected layer correspondsto a single point in the filtered projection vector. The number ofneurons may then correspond to a width of the corresponding image. Afterthis point-wise fully-connected layer, rotation and summation operationsare applied to simulate the analytical FBP method. Bilinearinterpolation may be used for rotating images. In one nonlimitingexample, C may be empirically set as 23, allowing the network to learnmultiple mappings from the sinogram domain to the image domain. Thevalue of C can be understood as the number of branches. Differentview-angle may use different parameters. Although the proposedfiltration and back-projection parts all together learn a refined FBPmethod, streak artifacts may not be eliminated perfectly. An imagereconstructed by the back-projection part 228-2 may thus be provided tothe refinement portion 228-3 of Gen 1 for refinement.

The refinement portion 228-1 may correspond to a U-net with conveyingpaths and may be constructed with the ResNeXt structure. In onenonlimiting example, U-net may be configured to contain 4 down-samplingand 4 up-sampling layers. Each layer may have a stride of 2 and may befollowed by a rectified linear unit (ReLU). A 3×3 kernel may be includedin both convolutional and transpose-convolutional layers. The number ofkernels in each layer is 36. To maintain the tensors size, zero-paddingis used.

Gen 2 network 232 is configured to have a same structure as therefinement portion 228-3 in Gen 1. The input 231 to G2 is aconcatenation of FBP-result 227 and output 229 from G1. With the use ofG2, the network becomes deep. As a result, the benefits of deep learningcan be utilized in this direct mapping for CT image reconstruction.

In operation, similar to the deep learning CT image reconstructionsystem 102 of FIG. 1 , DNA CT image reconstruction system 202 isoptimized using the Wasserstein Generative Adversarial Network (WGAN)framework. As described herein, the DNA CT image reconstruction system202 includes three components: two generator networks: Gen 1 network 228and Gen 2 network 232, and a discriminator network 234. Gen 1 and Gen 2aim at reconstructing images directly from a batch of few-viewsinograms. The discriminator network 234 is configured to receive imagesfrom Gen 1 and Gen 2 and a ground-truth dataset, and intends todistinguish whether an image is real (i.e., is from the ground-truthdataset) or fake (i.e., is from G1 or G2). The networks are configuredto be optimized in the training process. If an optimized network D canhardly distinguish fake images from real images, then it is concludedthat generators G1 and G2 can fool discriminator D which is the goal ofGAN. The network D is configured to help to improve the texture of thefinal image and prevent over-smoothed issue from occurring.

Different from generative adversarial network (GAN), Wasserstein GAN(WGAN) replaces the cross-entropy loss function with the Wassersteindistance, improving the training stability during the training process,as described herein. In an embodiment, an objective function used duringtraining of the DNA CT image reconstruction system 202 includes theWasserstein distance as well as a gradient penalty term. The objectivefunction of the WGAN framework for the DNA CT image reconstructionsystem 202 may be expressed as:

$\begin{matrix}{\min\limits_{\theta_{G_{1}},\theta_{G_{2}}}\max\limits_{\theta_{D}}\left\{ {{{\mathbb{E}}_{S_{SV}}\left\lbrack {D\left( {G_{1}\left( S_{SV} \right)} \right)} \right\rbrack} - {{\mathbb{E}}_{I_{FV}}\left\lbrack {D\left( I_{FV} \right)} \right\rbrack} - \text{ }\left\lbrack {D\left( {G_{2}\left( I_{SV} \right)} \right)} \right\rbrack - {{\mathbb{E}}_{I_{FV}}\left\lbrack {D\left( I_{FV} \right)} \right\rbrack} + {{\lambda\mathbb{E}}_{\overset{\_}{I}}\left\lbrack \left( {{{\nabla\left( \overset{\_}{I} \right)}}_{2} - 1} \right)^{2} \right\rbrack}} \right\}} & \left( {2B} \right)\end{matrix}$where S_(SV), I_(SV)=G_(I)(S_(SV)), I_(FV) represent a sparse-viewsinogram, an image reconstructed by Gen 1 from a sparse-view sinogramand the ground-truth image reconstructed from the full-view projectiondata, respectively. Similar to Eq. 2A, terms of the form

_(a)[b] in Eq. 2B denote an expectation of b as a function of a. θ_(G) ₁, θ_(G) ₂ and θ_(D) represent the trainable parameters of Gen 1 network228, Gen 2 network 232 and Discriminator network 234, respectively. Īrepresents images between fake (from G1 or G2) and real (from theground-truth dataset) images. ∇(Ī) denotes the gradient of D withrespect to Ī. The parameter λ balances the Wasserstein distance termsand gradient penalty terms. G1, G2 and D may be updated iteratively.

The objective function for optimizing the generator networks, Gen 1 andGen 2, may include the mean square error (MSE), structural similarityindex (SSIM) and adversarial loss. MSE is a popular choice for denoisingapplications, which effectively suppresses the background noise butcould result in over-smoothed images. Generally, MSE may be insensitiveto image texture since it assumes background noise is white Gaussiannoise and is independent of local image features. The formula of MSEloss (L₂) is expressed as Eq. 3, as described herein, where N_(b), W andH denote the number of batches, image width and image heightrespectively. Y_(i) and X_(i) represent ground-truth image and imagereconstructed by generator networks (G1 or G2), respectively.

To compensate for the disadvantages of MSE and acquire visually betterimages, SSIM is introduced in the objective function. The SSIM formulamay be expressed as Eq. 4, as described herein. The structural loss maythen be expressed as Eq. 5, as described herein.

The adversarial loss aims to assist the generators 228, 232, producingsharp images that are indistinguishable by the discriminator network234. Referring to Eq. 2B, adversarial loss for Gen 1 may be expressedas:L _(al) ⁽¹⁾=−

_(S) _(SV) [D(G ₁(S _(SV)))]  (6A)and adversarial loss for G2 is expressed as:L _(al) ⁽²⁾=−

_(S) _(SV) [D(G ₂(I _(SV)))]  (6B)

It may be appreciated that solving the few-view CT image reconstructionis similar to solving a set of linear equations when the number ofequations is not sufficient to perfectly resolve all the unknowns. DNACT image reconstruction system 202 is configured to estimate the unknownby combining the information from the existing equations and theknowledge contained in the big data. MSE between the original sinogramand the synthesized sinogram from a reconstructed image (from Gen 1 orGen 2) may be included as part of the objective function, which may bewritten as:

$\begin{matrix}{L_{2}^{sino} = {\frac{1}{N_{b} \cdot V \cdot H}{\sum\limits_{i = 1}^{N_{b}}{{Y_{i}^{sino} - {X_{i}^{sino}_{2}^{2}}}}}}} & \left( {3B} \right)\end{matrix}$where N_(b), V, H denote the number of batches, number of views andsinogram height, respectively. Y_(i) ^(sino) represents the originalsinogram and X_(i) ^(sino) represents sinogram from a reconstructedimage (from Gen 1 or Gen 2).

Both generator networks, Gen 1, Gen 2 may be updated at the same time.The overall objective function of two generators, e.g., generatornetworks 228, 232, may then be written as:

$\begin{matrix}{\min\limits_{\theta_{G_{1}},\theta_{G_{2}}}\text{ }\left\lbrack {{\lambda_{Q} \cdot \left( {L_{al}^{(1)} + L_{al}^{(2)}} \right)} + {\lambda_{P} \cdot \left( {L_{sl}^{(1)} + L_{sl}^{(2)}} \right)} + {\lambda_{R} \cdot \left( {L_{2}^{{sino}(1)} + L_{2}^{{sino}(2)}} \right)} + L_{2}^{(2)} + L_{2}^{(1)}} \right\rbrack} & \left( {7B} \right)\end{matrix}$where the superscripts (1) and (2) indicate that the term is formeasurements between ground-truth images and results reconstructed by G1and G2, respectively. λ_(Q), λ_(P) and λ_(R) are hyper-parameters usedto balance different loss functions.

The discriminator network 234 is configured to receive inputs from G1and G2, and the ground-truth dataset, and to try to distinguish whethereach input is real or fake. In one nonlimiting example, thediscriminator network 234 may include 6 convolutional layers with 64,64, 128, 128, 256, 256 filters and followed by 2 fully-connected layerswith numbers of neurons 1,024 and 1, respectively. The leaky ReLUactivation function may be used after each layer with a slope of 0.2,for example, in the negative part. A 3×3 kernel and zero-padding areused for all the convolutional layers, with stride equal 1 for oddlayers and stride equal 2 for even layers.

Thus, a dual network architecture CT image reconstruction system,consistent with the present disclosure, may include a plurality ofgenerator networks and a discriminator network. The generator networksand discriminator network may be trained, adversarially, using a WGANframework, as described herein. The DNA CT image reconstruction systemmay then be configured to receive CT scanner projection data (i.e.,sinograms), to filter the received projection data and to generate acorresponding image. The CT image reconstruction process of thegenerator networks is learned in a point-wise manner that facilitatesconstraining a memory burden. In some embodiments, the generatornetwork(s) and discriminator network may be pre-trained using, forexample, ImageNet data.

FIG. 3 is a flowchart 300 of deep learning CT image reconstructiontraining operations according to various embodiments of the presentdisclosure. In particular, the flowchart 300 illustrates training a deeplearning CT image reconstruction system to reconstruct an image from afew-view sinogram. The operations may be performed, for example, by deeplearning CT image reconstruction system 102 (e.g., preprocessing module124, generator network 126, and/or discriminator network 128) of FIG. 1.

In some embodiments, operations may include operation 302. Operation 302includes learning an initial filtered back-projection operation usingimage data from an image database that includes a plurality of images.For example, the image database may correspond to ImageNet. Operation304 may include receiving projection data (i.e., an input sinogram). Aramp filter may be applied to the input sinogram to yield a filteredsinogram at operation 306. The filtered sinogram may be received by afirst generator network at operation 308. Operation 310 may includelearning a filtered back-projection operation. A first reconstructedimage corresponding to the input sinogram may be provided as output atoperation 312. Operation 314 may include determining, by a discriminatornetwork, whether a received image corresponds to the first reconstructedimage or a corresponding ground truth image. The generator network andthe discriminator network correspond to a Wasserstein generativeadversarial network (WGAN). The WGAN is optimized using an objectivefunction based, at least in part, on a Wasserstein distance and based,at least in part, on a gradient penalty.

Thus, a deep learning CT image reconstruction system may be trained forfew-view CT image reconstruction.

FIG. 4 is a flow chart 400 of dual network architecture (DNA) CT imagereconstruction system training operations according to variousembodiments of the present disclosure. In particular, the flowchart 400illustrates training a DNA CT image reconstruction system to reconstructan image from a few-view sinogram. The operations may be performed, forexample, by DNA CT image reconstruction system 202 (e.g., preprocessingmodule 224, filtered back projection (FBP) module 226, first generatornetwork (Gen 1) 228, intermediate processing module 230, secondgenerator network (Gen 2) 232, and/or discriminator network 234) of FIG.2 .

In some embodiments, operations may include operation 402. Operation 402includes learning an initial filtered back-projection operation usingimage data from an image database that includes a plurality of images.For example, the image database may correspond to ImageNet. Operation404 may include receiving projection data (i.e., an input sinogram). Aramp filter may be applied to an input sinogram to yield a filteredsinogram at operation 406. The input sinogram may be processed by afiltered back projection module to yield a filtered back projectionresult at operation 408. The filtered sinogram may be received by afirst generator network at operation 410. Operation 412 may includelearning a filtered back-projection operation by the first generatornetwork. A first reconstructed image corresponding to the input sinogrammay be provided as an intermediate output at operation 414. The firstreconstructed image and a filtered back projection result may beconcatenated at operation 416. Operation 418 may include refining aconcatenation result by a second generator network. Operation 420 mayinclude determining, by a discriminator network, whether a receivedimage corresponds to the first reconstructed image, the secondreconstructed image or a corresponding ground truth image. The generatornetworks and the discriminator network correspond to a Wassersteingenerative adversarial network (WGAN). The WGAN is optimized using anobjective function based, at least in part, on a Wasserstein distanceand based, at least in part, on a gradient penalty.

Thus, a DNA CT image reconstruction system may be trained for few-viewCT image reconstruction.

As used in any embodiment herein, the terms “logic” and/or “module” mayrefer to an app, software, firmware and/or circuitry configured toperform any of the aforementioned operations. Software may be embodiedas a software package, code, instructions, instruction sets and/or datarecorded on non-transitory computer readable storage medium. Firmwaremay be embodied as code, instructions or instruction sets and/or datathat are hard-coded (e.g., nonvolatile) in memory devices.

“Circuitry”, as used in any embodiment herein, may include, for example,singly or in any combination, hardwired circuitry, programmablecircuitry such as computer processors comprising one or more individualinstruction processing cores, state machine circuitry, and/or firmwarethat stores instructions executed by programmable circuitry. The logicand/or module may, collectively or individually, be embodied ascircuitry that forms part of a larger system, for example, an integratedcircuit (IC), an application-specific integrated circuit (ASIC), asystem on-chip (SoC), desktop computers, laptop computers, tabletcomputers, servers, smart phones, etc.

The foregoing provides example system architectures and methodologies,however, modifications to the present disclosure are possible. Theprocessor 110 may include one or more processing units and may beconfigured to perform operations of one or more circuitries, modulesand/or artificial neural networks. Processing units may include, but arenot limited to, general-purpose processing units, graphical processingunits, parallel processing units, etc.

Memory 112 may include one or more of the following types of memory:semiconductor firmware memory, programmable memory, non-volatile memory,read only memory, electrically programmable memory, random accessmemory, flash memory, magnetic disk memory, and/or optical disk memory.Either additionally or alternatively system memory may include otherand/or later-developed types of computer-readable memory.

Embodiments of the operations described herein may be implemented in acomputer-readable storage device having stored thereon instructions thatwhen executed by one or more processors perform the methods. Theprocessor may include, for example, a processing unit and/orprogrammable circuitry. The storage device may include a machinereadable storage device including any type of tangible, non-transitorystorage device, for example, any type of disk including floppy disks,optical disks, compact disk read-only memories (CD-ROMs), compact diskrewritables (CD-RWs), and magneto-optical disks, semiconductor devicessuch as read-only memories (ROMs), random access memories (RAMs) such asdynamic and static RAMs, erasable programmable read-only memories(EPROMs), electrically erasable programmable read-only memories(EEPROMs), flash memories, magnetic or optical cards, or any type ofstorage devices suitable for storing electronic instructions.

The terms and expressions which have been employed herein are used asterms of description and not of limitation, and there is no intention,in the use of such terms and expressions, of excluding any equivalentsof the features shown and described (or portions thereof), and it isrecognized that various modifications are possible within the scope ofthe claims. Accordingly, the claims are intended to cover all suchequivalents.

Various features, aspects, and embodiments have been described herein.The features, aspects, and embodiments are susceptible to combinationwith one another as well as to variation and modification, as will beunderstood by those having skill in the art. The present disclosureshould, therefore, be considered to encompass such combinations,variations, and modifications.

What is claimed is:
 1. A few-view computed tomography (CT) imagereconstruction system, the system comprising: a preprocessing moduleconfigured to apply a ramp filter to an input sinogram to yield afiltered sinogram; a first generator network configured to receive thefiltered sinogram, to learn a filtered back-projection operation and toprovide a first reconstructed image as an output, the firstreconstructed image corresponding to the input sinogram; and adiscriminator network configured to determine whether a received imagecorresponds to the first reconstructed image or a corresponding groundtruth image, the first generator network and the discriminator networkcorresponding to a Wasserstein generative adversarial network (WGAN),the WGAN optimized using an objective function based, at least in part,on a Wasserstein distance and based, at least in part, on a gradientpenalty.
 2. The system of claim 1, further comprising a second generatornetwork configured to receive a concatenation of the first reconstructedimage and a filtered back-projection of the input sinogram and toprovide a second reconstructed image, the discriminator network furtherconfigured to determine whether the received image corresponds to thesecond reconstructed image.
 3. The system of claim 2, wherein the secondgenerator network corresponds to a refinement portion.
 4. The system ofclaim 2, further comprising a filtered back projection module configuredto receive the input sinogram and to provide the filteredback-projection of the input sinogram.
 5. The system of claim 1, whereinthe first generator network is configured to learn the filteredback-projection operation in a point-wise manner.
 6. The system of claim1, wherein the first generator network comprises a filtration portion, aback-projection portion, and a refinement portion.
 7. The system ofclaim 1, wherein the WGAN is trained, initially, using image data froman image database comprising a plurality of images.
 8. The system ofclaim 1, wherein the first generator network is configured toreconstruct the first reconstructed image using O(C×N×N,) parameters,where N is a dimension of the first reconstructed image, N, is a numberof projections, and C is an adjustable hyper-parameter in the range of 1to N.
 9. A method for few-view computed tomography (CT) imagereconstruction, the method comprising: applying, by a preprocessingmodule, a ramp filter to an input sinogram to yield a filtered sinogram;receiving, by a first generator network, the filtered sinogram;learning, by the first generator network, a filtered back-projectionoperation; providing, by the first generator network, a firstreconstructed image as an output, the first reconstructed imagecorresponding to the input sinogram; and determining, by a discriminatornetwork, whether a received image corresponds to the first reconstructedimage or a corresponding ground truth image, the first generator networkand the discriminator network corresponding to a Wasserstein generativeadversarial network (WGAN), the WGAN optimized using an objectivefunction based, at least in part, on a Wasserstein distance and based,at least in part, on a gradient penalty.
 10. The method of claim 9,further comprising: receiving, by a second generator network, aconcatenation of the first reconstructed image and a filteredback-projection of the input sinogram; providing, by the secondgenerator network, a second reconstructed image; and determining, by thediscriminator network, whether the received image corresponds to thesecond reconstructed image.
 11. The method of claim 10, furthercomprising receiving, by a filtered back projection module, the inputsinogram and providing, by the filtered back projection module, thefiltered back-projection of the input sinogram.
 12. The method of claim9, wherein the first generator network is configured to learn thefiltered back-projection operation in a point-wise manner.
 13. Themethod of claim 9, wherein the first generator network comprises afiltration portion, a back-projection portion, and a refinement portion.14. The method of claim 9, further comprising learning, by the firstgenerator network, an initial filtered back-projection operation usingimage data from an image database comprising a plurality of images. 15.The method of claim 9, wherein the first generator network is configuredto reconstruct the first reconstructed image using O(C×N×N,) parameters,where N is a dimension of the first reconstructed image, N, is a numberof projections, and C is an adjustable hyper-parameter in the range of 1to N.
 16. A computer readable storage device having stored thereoninstructions configured for a few-view computed tomography (CT) imagereconstruction, the instructions that when executed by one or moreprocessors result in the following operations comprising: applying aramp filter to an input sinogram to yield a filtered sinogram; receivingthe filtered sinogram; learning a filtered back-projection operation;providing a first reconstructed image as an output, the firstreconstructed image corresponding to the input sinogram; and determiningwhether a received image corresponds to the first reconstructed image ora corresponding ground truth image, the operations corresponding to aWasserstein generative adversarial network (WGAN), the WGAN optimizedusing an objective function based, at least in part, on a Wassersteindistance and based, at least in part, on a gradient penalty.
 17. Thedevice of claim 16, wherein the instructions that when executed by theone or more processors result in the following additional operationscomprising: receiving a concatenation of the first reconstructed imageand a filtered back-projection of the input sinogram; providing a secondreconstructed image; and determining whether the received imagecorresponds to the second reconstructed image.
 18. The device of claim16, wherein the filtered back-projection operation is learned in apoint-wise manner.
 19. The device of claim 16, wherein the instructionsthat when executed by the one or more processors result in the followingadditional operations comprising learning an initial filteredback-projection operation using image data from an image databasecomprising a plurality of images.
 20. The device of claim 16, whereinthe first reconstructed image is reconstructed using O(C×N×N,)parameters, where N is a dimension of the first reconstructed image, N,is a number of projections, and C is an adjustable hyper-parameter inthe range of 1 to N.